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An atmospheric internal gravity wave is an internal gravity wave propagating through a stably stratified atmosphere. Flow across mountains can generate nearly stationary waves whose vertical propagation is controlled by the wind and buoyancy-frequency profiles.

The portion of a fluid system's potential energy that can be released by admissible rearrangement relative to a stable reference state. In a uniformly stratified incompressible fluid, the quadratic density is $A={\rho }_0{N}^2{\xi }_z^2/2={\rho }_0{b}^2/(2N^2)$, where ${\xi }_z$ is vertical fluid displacement and $b$ the buoyancy perturbation. It is positive for stable stratification and measures the restoring energy of internal gravity waves.

A moving boundary excites an internal gravity wave through its kinematic boundary condition. For wavenumber $k$ and angular frequency $\omega$, the vertical velocity field satisfies $W^{\prime \prime }+k^2(N^2/{\omega }^2-1)W=0$. The radiation condition selects outward group velocity; when $\omega >N$, the bounded response is an evanescent wave.

In uniform positive flow $U$, upward stationary internal gravity waves have laboratory group velocity $U({\cos }^2\theta ,\sin \theta \cos \theta )$. Packets emitted by a localized source at startup lie on $(x-Ut/2{)}^2+z^2=(Ut/2{)}^2$, with later emissions inside. This envelope belongs to the permanent stationary component, not every possible startup transient.

For a two-dimensional internal gravity wave with phase $kx+\int m(z)dz-\omega t$, a constant-phase line at fixed time has slope $dz/dx=-k/m(z)$. Its normal spacing from the next crest is $2\pi /(k^2+m^2{)}^{1/2}$; the wavevector $(k,m)$ is normal to the line. The intrinsic energy ray is tangent to these phase lines, while mean-flow advection changes the laboratory ray direction.

A critical level is a height at which the intrinsic frequency of an internal gravity wave vanishes. For finite positive buoyancy frequency, its local vertical wavenumber diverges as $\hat{\omega }\to 0$. The inviscid Taylor–Goldstein equation is singular there; viscosity, diffusion or nonlinear wave breaking can regularize the resulting small scales.

A level where the squared vertical wavenumber of an internal gravity wave touches zero quadratically without changing sign. The ordinary WKB approximation fails there; unlike a simple sign-changing turning point, this does not itself provide an evanescent region on one side. Finite-wavelength analysis determines how solutions connect across or meet a boundary at the level.

For an internal gravity wave ray at angle $\theta$ to the horizontal reflecting from a slope at angle $\alpha$, conservation of tangential wavenumber givesAt a subcritical internal-wave reflection, $\gamma >1$: the reflected wavelength shortens and its wavelength-averaged energy density increases by ${\gamma }^2$.

A gravity wave is a fluid disturbance restored by gravity or buoyancy. A surface gravity wave displaces a free surface, while an internal gravity wave displaces density surfaces within a stratified fluid. Rotation can additionally contribute to restoration, giving an inertia-gravity wave.

Nonlinear disruption of an internal gravity wave, for example when large displacement gradients overturn the stable density arrangement or strong shear becomes unstable. The resulting turbulence dissipates wave energy and mixes the stratification, limiting ideal geometric focusing.

For a propagating, non-cutoff two-dimensional internal gravity wave proportional to $e^{i(kx+mz-\omega t)}$, incompressibility gives $U=-mW/k$. Thus the oscillatory velocity field is perpendicular to the wavevector and parallel to the group velocity, whereas the phase velocity is parallel to the wavevector.

Let an internal gravity wave energy ray make angle $\theta$ with the horizontal, so that $\sin \theta =\omega /N$. A boundary of local slope magnitude $s$ is subcritical, critical, or supercritical according as $s$ is smaller than, equal to, or larger than $\tan \theta$. At criticality the inviscid reflected wavelength tends to zero.

At a step in buoyancy frequency without a jump in background mass density, vertical velocity and fluid pressure are continuous. For a fixed horizontal wavenumber and angular frequency, these are continuity of $W$ and $W^{\prime }$. They determine reflection and transmission between propagating internal gravity waves and evanescent waves.

Across a horizontal material interface with continuous background mass density but a velocity jump, an internal gravity wave preserves laboratory frequency and horizontal wavenumber. The matching conditions are continuity of displacement $w/\hat{\omega }$ and pressure, not continuity of $w$. For wave amplitudes and upward phase-line angles measured from the vertical ${\theta }_1,{\theta }_2$, define $r={\hat{\omega }}_2/{\hat{\omega }}_1$. Then the transmission coefficient isThe displacement condition is $A_i+A_r=A_t\cos {\theta }_2/(r\cos {\theta }_1)$, while pressure continuity gives $A_i-A_r=r{A}_t\sin {\theta }_2/\sin {\theta }_1$. Adding eliminates the reflected amplitude and gives the coefficient. It assumes nonzero intrinsic frequencies and propagating outgoing branches.